A mass ‘m’ of fluid at temperature T1 is mixed with an equal amount of the same fluid at temperature T2. Prove that the resultant change of entropy of the universe is: [2 mc ln (T1 + T2) / 2]/[(T1 T2)1/2]and also prove that it is always positive.

Mean temperature of the mixture = (T1 + T2) / 2

Thus change in entropy is given by:
∆S = S2 – S1)

= mc (T1 + T2)/2 T1 (dT / T) – mc T2 T1 + T2)/2 (dT / T)

= mc ln (T1 + T2)/2 T1) –
mc ln (2 T2) / (T1 + T2)
= mc ln (T1 + T2)/2 T1) +
mc ln (T1 + T2) /2 T2)

= mc ln (T1 + T2) 2 / 4 T1 T2

= mc ln [(T1 + T2) / 2 (T1 T2) 1/2] 2

= 2 mc ln [(T1 + T2) / 2 (T1 T2) 1/2]

= 2 mc ln [(T1 + T2) / 2 ]/[ (T1 T2)1/2]

Hence, Resultant change of entropy of universe is:
2 mc ln [(T1 + T2) / 2]/[ (T1 T2)1/2]

The arithmetic mean (T1 + T2) / 2 is greater than the geometric mean (T1 T2) 1/2

Therefore, ln [(T1 + T2) / 2]/[ (T1 T2)1/2 ]

is always positive. Hence the entropy of the universe increases.

Category: Second Law of Thermodynamics

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